L(s) = 1 | + 5-s − 5.18·7-s + 5.53·11-s + 1.78·13-s − 2.22·17-s − 8.22·19-s + 23-s + 25-s + 7.06·29-s − 2.06·31-s − 5.18·35-s − 5.02·37-s + 11.4·41-s − 8.62·43-s + 1.37·47-s + 19.9·49-s − 4.46·53-s + 5.53·55-s + 11.1·59-s + 7.81·61-s + 1.78·65-s − 1.96·67-s − 8.51·71-s − 10.7·73-s − 28.6·77-s − 14.0·79-s + 6.83·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.96·7-s + 1.66·11-s + 0.496·13-s − 0.539·17-s − 1.88·19-s + 0.208·23-s + 0.200·25-s + 1.31·29-s − 0.371·31-s − 0.876·35-s − 0.826·37-s + 1.79·41-s − 1.31·43-s + 0.200·47-s + 2.84·49-s − 0.613·53-s + 0.745·55-s + 1.45·59-s + 1.00·61-s + 0.221·65-s − 0.240·67-s − 1.01·71-s − 1.25·73-s − 3.26·77-s − 1.57·79-s + 0.750·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 5.18T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 8.22T + 19T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.81T + 61T^{2} \) |
| 67 | \( 1 + 1.96T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 4.19T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.04714588915795881407280245033, −6.65857248324928868408804049605, −6.28735572692018192186513769162, −5.67627244858319396215258474341, −4.35627562260940883352825355368, −3.93973137889485042276198923858, −3.11142428144872488647311875728, −2.31085032481937241033234797353, −1.21152083180771539822521474062, 0,
1.21152083180771539822521474062, 2.31085032481937241033234797353, 3.11142428144872488647311875728, 3.93973137889485042276198923858, 4.35627562260940883352825355368, 5.67627244858319396215258474341, 6.28735572692018192186513769162, 6.65857248324928868408804049605, 7.04714588915795881407280245033